Calculus: The Language of Change

Exercise Set 15.3
Geometry of Vector Addition

solutions by Randall Childs and David Theobald

See the labeled Mathematica notebook to solve Computer Exercises 1.

Where do these fancy graphs come from?

The graphs in this notebook were drawn with the aid of the special commands defined in several of the notebooks located under Chapter 15 in the MathmCTLC directory.  The Mathematica gymnastics used to define these special commands are too complicated to explain.  In fact, even the solutions author does not understand them.  Suffice it to say that they take a large set of simpler Mathematica graphics commands and combines them so that the user may draw useful pictures of vectors without crunching a lot of computer programming code.

Exercise 2

Use a diagram and explain that the sum X+Y of two 3-dimensional position vectors X and Y is the directed diagonal of a parrelogram with X and Y as two adjacent sides.

First draw the position vectors X and Y, (the blue and red ones, respectively).  The vector sum X + Y can be represented by drawing the (orange) copy of Y with its tail at the tip of X.  The vector sum Y + X can be represented by drawing the (green) copy of X with its tail at the tip of Y.  The position vector X + Y and the position vector Y + X are the same vector (the black one).  This makes sense algebraically:

X  +  Y = (({{(Subscript[x, 1])}, {({{(Subscript[x, 2])}, {(Subscript[x, 3])}})}})) + (({{(Subscript[y, 1])}, {({{(Subscript[y, 2])}, {(Subscript[y, 3])}})}})) = (({{(Subscript[x, 1]) +  (Subscript[y, 1])}, {({{(Subscript[x, 2]) +  (Subscript[y, 2])}, {(Subscript[x, 3]) +  (Subscript[y, 3])}})}})) = (({{(Subscript[y, 1]) +  (Subscript[x, 1])}, {({{(Subscript[y, 2]) +  (Subscript[x, 2])}, {(Subscript[y, 3]) +  (Subscript[x, 3])}})}})) = (({{(Subscript[y, 1])}, {({{(Subscript[y, 2])}, {(Subscript[y, 3])}})}})) + (({{(Subscript[x, 1])}, {({{(Subscript[x, 2])}, {(Subscript[x, 3])}})}})) = Y  +  X

In other words, vector addition is commutative.

Now from the diagram it is clear that X, Y, and their copies form a quadrilateral whose opposite sides are parallel, i.e., a parallelogram.  Clearly, X and Y are adjacent sides, and the vector X + Y is a (directed) diagonal of this parallelogram.

See Example 15.19.

Exercise 3

What geometric property must X and Y share for the formula |X+Y| = |X| + |Y| to be true?

In the following diagram, the red arrow is the copy of the vector Y with the tail placed at the tip of X.

The blue, red, and black arrows trace the sides of a triangle.  There is a geometric condition known as the Triangle Inequality which states the length of any side of a triangle is less than the sum of the lengths of the other two sides.  This is just another way of saying that the shortest distance between two points is a straight line.

So the triangle inequality gives the following relationship between the lengths of X, Y, and X+Y:

|X| + |Y| > |X+Y|

So long as X, 'Y', and X+Y form a triangle, this inequality will hold.  The reason they form a triangle is because X and Y point in different directions.  If they point in the same direction, then the following equation clearly holds:

|X| + |Y| = |X+Y|

What if X and Y pointed in diametrically opposite directions?